^60 



Algebra g„ i =aS./3y(); j = aS.ya(); /; = /? 8. ^ >' ( ) ; / = /3S.ya(). 



" hp„i=l,S.{)A.lJJ,-l,S.{)A.l,lJ,;j = -l,S.()A.lJ,l,- 



k = ~-l^S.{ ) A. l^lji — 1-, S.( )A. IJrJ^ ; 1=1, S.( )A. ^Ji/j ; 



" bkg,!-^/^ S. ( ) A. U3/4 -?2 ^- ( ) A. l,Uh + /3 S. ( ) A. /,/i/, ; i^- 

 Zi S. i ) A. LJJ., + /2 S. ( ) A. /,/i/, ; k = l, S. A. /,/,/, ; 

 l = -l,^.()A.I,lJ,; m = -l,S.{ )A.l,lJ,; n = -l,8. 

 {) A.l,l,h. 

 hmQ,i = a S. p y { );J = a S. y a i) • k= a 8. a 13 {); 1 = fi8. fiy {) ;m = 

 i3S.ya{)- n = i3Rap{ ). 

 These examples can be used to illustrate the general theorems. For example: 

 " Every group of linear vector operators contains at least one idempotent or one 

 nilpotenl expresssion." 



The group b mg contains the idempotents 



«S./3j(), /?S. ya(), a S. ,8 y ( ) + /3 S. y a ( ). 

 The group b pg contains only nilpotents. 



" When an algebra contains an idempotent expression it may be assicmed as the 

 basis and tJie remaining expressions are then divisihle into four classes." 



In b mg if we assume a 8. (3 y { ) as the idempotent then the units are, with 

 reference to the basis, 



idemfaciend, idemfacient, a S. (3 y { ) ; 

 nilfaoiend, idemfacient, /? S. /? y ( ) ; 

 idemfaciend, nilfacient, a S. y a ( ), and a S. a f3 () ; 

 nilfaciend, nilfacient, /? S. y a ( ), and /? S. « /3 ( ). 



" The fourth class are subject to independent investigation." 



"If the first class comprises any units except the basis, there is, besides the basis, another 

 idempotent expression or a nilpotent expression, and we may free the class from this, when 

 idempotent, by writing for the basis the difference between the two ; in this case expressions 

 may pass from idemfaciend to nilfaciend or from idemfacient to nilfacient, but not the 

 reverse." Thus, if we had taken for our basis in b mg « S. /? y ( ) + ^3 S. y « ( ) there 

 would have been only two classes, 



1 : a S. /3 y ( ) + /? S. y a ( ) ; (3 S. y a { ) ; a S. y a { ) ; f3 S. (3 y { ) ; 



2 : aS.a(3(); ^ S. a /? ( ). 



The second idempotent basis is easily seen to be (3 S. y a { ), and the diflerence 

 is a S. /3 y ( ), as before. And making this change of basis, f3S.y a ( ) and /S S. a /? ( ) 

 become fourth class, /'? S. /? y ( ) becomes second class, a S. y a ( ) becomes third class. 



" When there is no idempotent 6a.s'is, all expressions are nilpotent, and all porcers of 

 ^ach expression that do not vanish are independent. We may take any expression as the 



